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Q3:  Given an integer tuple xvec=(a, b, c, d) of any length, let
	sumsqs(xvec)
mean the "sum of squares" a²+b²+c²+d².


The LAGRANGE 4SQUARE THEOREM says that for each posint K, there
exist a natnum 4-tuple xvec [some entries might be zero] so that
	sumsqs(xvec) = K.

Part of the proof involves "melding" the polynomial a²+b²+c²+d²,
analogously to our melding of poly x²+y².

DEFINITION:  Given 4-tuples
    xvec := (x1, x2, x3, x4)  and    yvec := (y1, y2, y3, y4),
define the  "Lagrange meld"
	
1:	zvec := Lagmeld(xvec, yvec)

as shown on page NZM317, below the statement of Thm6.26.  Here,
z1 is the contents of the first pair of parentheses, on the
RHSide of the equality.  Etc.

Q3a:  Note that
	155 = 9²+7²+5²	and
	 57 = 2²+4²+6²+1² .
Please write  8835 [= 155*57] as a sum of four squares,
showing the interesting aspects of the lagmelding.


Q3b:  Do something interesting with lagmelding; make up an
interesting problem and then solve it.


Q3c:  Give a formal proof, from (1),  that

	sumsqs(zvec) = sumsqs(xvec)·sumsqs(yvec).

[NZM does not give a proof.]  You may want to break your
argument into a lemma or two.  

Aside for-the-Curious:  Note that x²+y² is the "norm" of the
complex number x+yi.  [By "norm" I mean here the square of
absolute value.]   Thus that "melding is sealed under
multiplication" is simply the statement that the abs-value of a
product [of complex numbers] is the product of their
abs-values. 

 It turns out that 

	a²+b²+c²+d²

is the norm of the "quaternion" a+bi+cj+dk.  [Of course, you don't
need to know this in order to do the problem.]

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